String periods in the order-preserving model

Gourdel, Garance; Kociumaka, Tomasz; Radoszewski, Jakub; Rytter, Wojciech; Shur, Arseny M.; Waleń, Tomasz

doi:10.1016/j.ic.2019.104463

Cited by 4 publications

(1 citation statement)

References 40 publications

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“…and the periodicity lemma states that if p and q are both such periods and p + q ≤ n + gcd(p, q) then gcd(p, q) is also a period [32]. This was generalised in a myriad of ways, for strings [17,49,74], partial words (words with don't cares) [11-13, 47, 50, 69, 70], Abelian periods [14,20], parametrized periods [6], order-preserving periods [42,64], approximate periods [2][3][4]. Now, a square can be defined as a fragment of length twice its period.…”

Section: Introductionmentioning

confidence: 99%

Optimal Square Detection Over General Alphabets

Ellert¹,

Gawrychowski²,

Gourdel³

2023

Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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Squares (fragments of the form xx, for some string x) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length n is square-free, that is, does not contain a fragment of such form. Main and Lorentz [J. Algorithms 1984] designed an O(n log n) time algorithm for this problem, and proved a matching lower bound assuming the so-called general alphabet, meaning that the algorithm is only allowed to check if two characters are equal. However, their lower bound also assumes that there are Ω(n) distinct symbols in the string. As an open question, they asked if there is a faster algorithm if one restricts the size of the alphabet. Crochemore [Theor. Comput. Sci. 1986] designed a linear-time algorithm for constant-size alphabets, and combined with more recent results his approach in fact implies such an algorithm for linearly-sortable alphabets. Very recently, Ellert and Fischer [ICALP 2021] significantly relaxed this assumption by designing a linear-time algorithm for general ordered alphabets, that is, assuming a linear order on the characters that permits constant time order comparisons. However, the open question of Main and Lorentz from 1984 remained unresolved for general (unordered) alphabets. In this paper, we show that testing square-freeness of a length-n string over general alphabet of size σ can be done with O(n log σ) comparisons, and cannot be done with o(n log σ) comparisons. We complement this result with an O(n log σ) time algorithm in the Word RAM model. Finally, we extend the algorithm to reporting all the runs (maximal repetitions) in the same complexity.

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Section: Introductionmentioning

confidence: 99%

Optimal Square Detection Over General Alphabets

Ellert¹,

Gawrychowski²,

Gourdel³

2023

Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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Computing Covers Under Substring Consistent Equivalence Relations

Kikuchi

Hendrian

Yoshinaka

et al. 2020

String Processing and Information Retrieval

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Order-Preserving Pattern Matching with Partition

Na,

Kim,

Kang

et al. 2024

Mathematics

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Order-preserving pattern matching, which considers the relative orders of strings, can be applied to time-series data analysis. To perform a more meaningful analysis of time-series data, approximate criteria for the order-isomorphism are necessary, considering diverse types of errors. In this paper, we introduce a novel approximation criterion for the order-isomorphism, called the partitioned order-isomorphism. We then propose an efficient O(n+sort(m))-time algorithm for the order-preserving pattern matching problem considering the criterion of partition. A comparative experiment demonstrates that the proposed algorithm is more effective than the exact order-preserving pattern matching algorithm.

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String periods in the order-preserving model

Cited by 4 publications

References 40 publications

Optimal Square Detection Over General Alphabets

Optimal Square Detection Over General Alphabets

Computing Covers Under Substring Consistent Equivalence Relations

Order-Preserving Pattern Matching with Partition

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